# Which capital accumulation is right? $K_t = (1-\delta)K_{t-1} +I_t$ or $K_t = (1-\delta)K_{t-1}+I_{t-1}$?

In a lot of models with capital, I find different variants of capital accumulation formula as follows: $K_t = (1-\delta)K_{t-1} +I_t$ or

$K_t = (1-\delta)K_{t-1}+I_{t-1}$

Which one is more economics-sound formula?

Both are economically sound. The notation is just a question of convention. The reason behind the ambiguity is that capital is a stock and investment is a flow variable. You are looking at capital in two different instants. Investment happens during the time between the two instants and its index is either the starting or the ending instant.

This is a bit more complicated than it looks. The two are different but equivalent notational conventions, only if it is clear that they incorporate the same assumption of essence (that is rarely stated explicitly nowadays), i.e. that it takes only one period for investment to become part of capital and so productive.

Under this assumption the notational difference comes from what is the meaning that we assign to the index for the state variable, the capital.

Some models designate $K_t$ to mean "Capital stock at the beginning of period $t$. In such a case, for example the production function for period $t$ should include $K_t$. As for the law of motion of capital, since investment happens during period $t$, and augments the capital that was there at the beginning of the period, here the law of motion is usually written, for clarity,

$$K_{t+1} = (1-\delta) K_t + I_t$$ and lagging it, we obtain $K_t = (1-\delta)K_{t-1}+I_{t-1}$

Others designate $K_t$ to mean "Capital stock at the end of period $t$. With the same logic as before, the production function for period $t$ should include $K_{t-1}$ and the law of motion of capital should be written

$$K_{t} = (1-\delta)K_{t-1} + I_t$$

But assume that I want to create a model where investment takes two periods to become part of capital (i.e. productive), and I also want to use the second notational convention of the above. I will have to write the law of motion of capital as $$K_{t} = (1-\delta)K_{t-1} + I_{t-1}$$ which looks exactly like "capital at the beginning of the period plus investment productive in one period", but on the (totally) contrary, it is here meant to be translated "capital stock at the end of period $t$ (and hence available for production in period $t+1$), that is productively augmented by investment made during period $t-1$".

It would perhaps be better then to use the first notational convention and write

$$K_{t+1} = (1-\delta) K_t + I_{t-1}$$

But in general one has to carefully read the assumptions and the notational conventions of the model.

Both can be correct, depending on the timing of three events--namely production, investment, and depreciation.

• $K_t = (1-\delta)K_{t-1} +I_t$

This would correspond to a model where depreciation takes place at the end of yesterday, investment at the start of today, and production in the middle of today.

• $K_t = (1-\delta)K_{t-1}+I_{t-1}$

This would correspond to a model where depreciation and then investment take place at the end of yesterday, and production at the start of today.